Learning Communities in the Presence of Errors

Abstract

We study the problem of learning communities in the presence of modeling errors and give robust recovery algorithms for the Stochastic Block Model (SBM). This model, which is also known as the Planted Partition Model, is widely used for community detection and graph partitioning in various fields, including machine learning, statistics, and social sciences. Many algorithms exist for learning communities in the Stochastic Block Model, but they do not work well in the presence of errors. In this paper, we initiate the study of robust algorithms for partial recovery in SBM with modeling errors or noise. We consider graphs generated according to the Stochastic Block Model and then modified by an adversary. We allow two types of adversarial errors, Feige—Kilian or monotone errors, and edge outlier errors. Mossel, Neeman and Sly (STOC 2015) posed an open question about whether an almost exact recovery is possible when the adversary is allowed to add o(n) edges. Our work answers this question affirmatively even in the case of k>2 communities. We then show that our algorithms work not only when the instances come from SBM, but also work when the instances come from any distribution of graphs that is \varepsilon m close to SBM in the Kullback—Leibler divergence. This result also works in the presence of adversarial errors. Finally, we present almost tight lower bounds for two communities.

Related Material

@InProceedings{pmlr-v49-makarychev16,
title = {Learning Communities in the Presence of Errors},
author = {Konstantin Makarychev and Yury Makarychev and Aravindan Vijayaraghavan},
booktitle = {29th Annual Conference on Learning Theory},
pages = {1258--1291},
year = {2016},
editor = {Vitaly Feldman and Alexander Rakhlin and Ohad Shamir},
volume = {49},
series = {Proceedings of Machine Learning Research},
address = {Columbia University, New York, New York, USA},
month = {23--26 Jun},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v49/makarychev16.pdf},
url = {http://proceedings.mlr.press/v49/makarychev16.html},
abstract = {We study the problem of learning communities in the presence of modeling errors and give robust recovery algorithms for the Stochastic Block Model (SBM). This model, which is also known as the Planted Partition Model, is widely used for community detection and graph partitioning in various fields, including machine learning, statistics, and social sciences. Many algorithms exist for learning communities in the Stochastic Block Model, but they do not work well in the presence of errors. In this paper, we initiate the study of robust algorithms for partial recovery in SBM with modeling errors or noise. We consider graphs generated according to the Stochastic Block Model and then modified by an adversary. We allow two types of adversarial errors, Feige—Kilian or monotone errors, and edge outlier errors. Mossel, Neeman and Sly (STOC 2015) posed an open question about whether an almost exact recovery is possible when the adversary is allowed to add o(n) edges. Our work answers this question affirmatively even in the case of k>2 communities. We then show that our algorithms work not only when the instances come from SBM, but also work when the instances come from any distribution of graphs that is \varepsilon m close to SBM in the Kullback—Leibler divergence. This result also works in the presence of adversarial errors. Finally, we present almost tight lower bounds for two communities. }
}

%0 Conference Paper
%T Learning Communities in the Presence of Errors
%A Konstantin Makarychev
%A Yury Makarychev
%A Aravindan Vijayaraghavan
%B 29th Annual Conference on Learning Theory
%C Proceedings of Machine Learning Research
%D 2016
%E Vitaly Feldman
%E Alexander Rakhlin
%E Ohad Shamir
%F pmlr-v49-makarychev16
%I PMLR
%J Proceedings of Machine Learning Research
%P 1258--1291
%U http://proceedings.mlr.press
%V 49
%W PMLR
%X We study the problem of learning communities in the presence of modeling errors and give robust recovery algorithms for the Stochastic Block Model (SBM). This model, which is also known as the Planted Partition Model, is widely used for community detection and graph partitioning in various fields, including machine learning, statistics, and social sciences. Many algorithms exist for learning communities in the Stochastic Block Model, but they do not work well in the presence of errors. In this paper, we initiate the study of robust algorithms for partial recovery in SBM with modeling errors or noise. We consider graphs generated according to the Stochastic Block Model and then modified by an adversary. We allow two types of adversarial errors, Feige—Kilian or monotone errors, and edge outlier errors. Mossel, Neeman and Sly (STOC 2015) posed an open question about whether an almost exact recovery is possible when the adversary is allowed to add o(n) edges. Our work answers this question affirmatively even in the case of k>2 communities. We then show that our algorithms work not only when the instances come from SBM, but also work when the instances come from any distribution of graphs that is \varepsilon m close to SBM in the Kullback—Leibler divergence. This result also works in the presence of adversarial errors. Finally, we present almost tight lower bounds for two communities.

TY - CPAPER
TI - Learning Communities in the Presence of Errors
AU - Konstantin Makarychev
AU - Yury Makarychev
AU - Aravindan Vijayaraghavan
BT - 29th Annual Conference on Learning Theory
PY - 2016/06/06
DA - 2016/06/06
ED - Vitaly Feldman
ED - Alexander Rakhlin
ED - Ohad Shamir
ID - pmlr-v49-makarychev16
PB - PMLR
SP - 1258
DP - PMLR
EP - 1291
L1 - http://proceedings.mlr.press/v49/makarychev16.pdf
UR - http://proceedings.mlr.press/v49/makarychev16.html
AB - We study the problem of learning communities in the presence of modeling errors and give robust recovery algorithms for the Stochastic Block Model (SBM). This model, which is also known as the Planted Partition Model, is widely used for community detection and graph partitioning in various fields, including machine learning, statistics, and social sciences. Many algorithms exist for learning communities in the Stochastic Block Model, but they do not work well in the presence of errors. In this paper, we initiate the study of robust algorithms for partial recovery in SBM with modeling errors or noise. We consider graphs generated according to the Stochastic Block Model and then modified by an adversary. We allow two types of adversarial errors, Feige—Kilian or monotone errors, and edge outlier errors. Mossel, Neeman and Sly (STOC 2015) posed an open question about whether an almost exact recovery is possible when the adversary is allowed to add o(n) edges. Our work answers this question affirmatively even in the case of k>2 communities. We then show that our algorithms work not only when the instances come from SBM, but also work when the instances come from any distribution of graphs that is \varepsilon m close to SBM in the Kullback—Leibler divergence. This result also works in the presence of adversarial errors. Finally, we present almost tight lower bounds for two communities.
ER -